Finite Morse Index Solutions of a Nonlinear Schrödinger Equation

Acta Mathematica Sinica, English Series - We prove Liouville type theorems for stable and finite Morse index H loc 1 ∩ L loc ∞ solutions of the nonlinear Schrödinger equation...     

Solutions of the Discrete Nonlinear Schrödinger Equation with a Trap

Theoretical and Mathematical Physics - We obtain solutions of the discrete nonlinear Schrödinger equation with an impurity center in two ways. In the first of them, we construct the wave...     

Multiple Solutions for a Nonlinear Schrödinger Equation with Magnetic Fields

We study a nonlinear Schrödinger equation in presence of a magnetic field and relate the number of solutions with the topology of the set where the potential attains its minimum value. In the proofs we apply variational methods, penalization techniques and Ljusternik–Schnirelmann theory.     

On Multiwave Solutions of One Nonlinear Schrödinger Equation

Differential Equations - We consider a nonlinear Schrödinger equation arising in a number of physical problems. It is shown that when the real part is separated in this equation, there arises...     

Finite Difference Method for an Optimal Control Problem for a Nonlinear Time-dependent Schrödinger Equation

The finite difference method is applied to an optimal control problem for a system governed by a nonlinear Schrödinger equation with a complex coe?cient. The optimal control problem is discretized by the finite difference method, the error estimate for the finite difference scheme is established and the convergence of difference approximations of the optimal control according to the functional is proved.     

Blow-up of Solutions of the Cauchy Problem for a Nonlinear Schrödinger Evolution Equation

The solution of the Cauchy problem for a nonlinear Schrödinger evolution equation with certain initial data is proved to blow up in a finite time, which is estimated from above. Additionally, lower bounds for the blow-up rate are obtained in some norms.     

The Nonlinear Steepest Descent Method to Long-Time Asymptotics of the Coupled Nonlinear Schrödinger Equation

The Riemann–Hilbert problem for the coupled nonlinear Schrödinger equation is formulated on the basis of the corresponding \(3\times 3\) matrix spectral problem. Using the nonlinear steepest descent method, we obtain leading-order asymptotics for the Cauchy problem of the coupled nonlinear Schrödinger equation.     

Sign-changing Solutions for a Schrödinger Equation with Saturable Nonlinearity

We consider the nonlinear Schrödinger equation

$-\Delta{u} + V (x)u = K(x)u^3/(1 + u^2)$

in \({\mathbb {R}^N}\) , and assume that V and K are invariant under an orthogonal involution. Moreover, V and K converge to positive constants V _∞ and K _∞, as |x| → ∞. We present some results on the existence of a particular type of sign changing solution, which changes sign exactly once. The basic tool employed here is the Concentration–Compactness Principle and the interaction between translated solutions of the corresponding autonomous problem.

     

Existence of Solutions for a Quasilinear Schrödinger Equation with Potential Vanishing

Acta Mathematicae Applicatae Sinica, English Series - We study the following quasilinear Schrödinger equation $$ - \Delta u + V(x)u - \Delta ({u^2})u = K(x)g(u),\,\,\,\,\,\,\,\,x \in...     

Sobolev Stability of Plane Wave Solutions to the Nonlinear Schrödinger Equation

This article explores the questions of long time orbital stability in high order Sobolev norms of plane wave solutions to the NLSE in the defocusing case.     

Symmetries of the Discrete Nonlinear Schrödinger Equation

The Lie algebra L(h) of point symmetries of a discrete analogue of the nonlinear Schrödinger equation (NLS) is described. In the continuous limit, the discrete equation is transformed into the NLS, while the structure of the Lie algebra changes: a contraction occurs with the lattice spacing h as the contraction parameter. A five-dimensional subspace of L(h), generated by both point and generalized symmetries, transforms into the five-dimensional point symmetry algebra of the NLS.     

Finite Dimensional Behavior of Global Attractors for Weakly Damped and Forced KdV Equations Coupling With Nonlinear Schrodinger Equation

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Global Solutions of Modified One-Dimensional Schrödinger Equation

In this paper, we consider the modified one-dimensional Schrödinger equation:$(D_t-F(D))u=λ|u|^2u,$where F(ξ) is a second order constant coefficients classical elliptic symbol, and with smooth initial datum of size $ε≪1$. We prove that the solution is global-in-time, combining the vector fields method and a semiclassical analysis method introduced by Delort. Moreover, we get a one term asymptotic expansion for $u$when $t→+∞$.     

Finite Time Extinction by Nonlinear Damping for the Schrödinger Equation

We consider the Schrödinger equation on a compact manifold, in the presence of a nonlinear damping term, which is homogeneous and sublinear. For initial data in the energy space, we construct a weak solution, defined for all positive time, which is shown to be unique. In the one-dimensional case, we show that it becomes zero in finite time. In the two and three-dimensional cases, we prove the same result under the assumption of extra regularity on the initial datum.     

Nonlinear Evolutionary Schrödinger Equation in a Two-Dimensional Domain

Theoretical and Mathematical Physics - We consider a mixed problem for a nonlinear evolutionary Schrödinger equation in a two-dimensional domain and study the smoothness of solutions and their...     

Breathers for the Discrete Nonlinear Schrödinger Equation with Nonlinear Hopping

We discuss the existence of breathers and lower bounds on their power, in nonlinear Schrödinger lattices with nonlinear hopping. Our methods extend from a simple variational approach to fixed-point arguments, deriving lower bounds for the power which can serve as a threshold for the existence of breather solutions. Qualitatively, the theoretical results justify non-existence of breathers below the prescribed lower bounds of the power which depend on the dimension, the parameters of the lattice as well as of the frequency of breathers. In the case of supercritical power nonlinearities we investigate the interplay of these estimates with the optimal constant of the discrete interpolation inequality. Improvements of the general estimates, taking into account the localization of the true breather solutions are derived. Numerical studies in the one-dimensional lattice corroborate the theoretical bounds and illustrate that in certain parameter regimes of physical significance, the estimates can serve as accurate predictors of the breather power and its dependence on the various system parameters.     

Collapse Rate of Solutions of the Cauchy Problem for the Nonlinear Schrödinger Equation

We prove that solutions of the Cauchy problem for the nonlinear Schrödinger equation with certain initial data collapse in a finite time, whose exact value we estimate from above. We obtain an estimate from below for the solution collapse rate in certain norms.